I can't help you with that unless you tell me how your textbook defines [itex]H'[/itex], [itex]E_n^1[/itex] and [itex]\psi_n^0[/itex]. I am prepared to guess that you have
[tex]E_n = E_n^0 + \epsilon E_n^1 + O(\epsilon^2)\\
\psi_n = \psi_n^0 +\epsilon\psi_n^1 + O(\epsilon^2)[/tex]
but I'm not prepared to guess what [itex]H'[/itex] might be. However I suspect the method is to substitute the above into the Schrodinger equation to get
[tex]
(E_n^0 + \epsilon E_n^1)(\psi_n^0 + \epsilon \psi_n^1) = -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} (\psi_n^0 + \epsilon\psi_n^1) + \frac{kx^2}{2}(1 + \epsilon)(\psi_n^0 +\epsilon\psi_n^1)
[/tex]
and then require that the coefficients of [itex]\epsilon^0[/itex] and [itex]\epsilon^1[/itex] should vanish. At some stage you may want to take an inner product with [itex]\langle\psi_n^0|[/itex], and recall that
[tex]H_0 = -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{kx^2}{2}[/tex]
is self-adjoint ([itex]\langle f | H_0 | g \rangle = \langle g | H_0 | f \rangle[/itex] for all [itex]f[/itex] and [itex]g[/itex]).